Students will study patterns in products and quotients when multiplying or dividing by powers of 10. Students will: 
- denote powers of 10 using whole number exponents. 
- explain patterns in the number of zeros in a product or quotient when multiplying or dividing by a power of ten. 
- explain how to insert a decimal point into a product or quotient when multiplying or dividing a decimal number by a power of ten. 

- How can mathematics help to quantify, compare, depict, and model numbers?
- How can mathematics help us communicate more effectively?
- How are relationships represented mathematically?
- What makes a tool and/or strategy suitable for a certain task?
- How may patterns be used to describe mathematical relationships?

- Decimal Place Value: A place value to the right of the decimal point in a number. The base for a decimal place value is less than 1. Each place value after the decimal point is \(1 \over 10\) of the place value to its left.
- Expanded Form: A method for representing a base-ten number as the sum of its parts, each represented by its base value multiplied by a power of ten.
- Hundredths Place: The place value two places to the right of the decimal point in a base-ten number. The digit located in this place represents a fractional part out of one hundred.
- Tenths Place: The place value on the right of the decimal point in a base-ten number. The digit in this place represents a fractional part out of ten.
- Thousandths Place: The place value three places to the right of the decimal point in a base-ten number. The digit in this place represents a fractional part out of one thousand.
- Exponent: A number used to show the number of times a base value should be multiplied repeatedly by itself.

- student copies of Vocabulary Journal pages (M-5-5-1_Vocabulary Journal)
- calculators for students or one that displays on the overhead or interactive whiteboard
- teacher set of base-ten blocks or use virtual set on the interactive whiteboard
- student sets of base-ten blocks or use paper base-ten blocks, either cut apart ahead or plan time for students to do it (M-5-5-1_Paper Base-Ten Models)
- sticky notes (about one per student)
- student copies of the Power of Ten practice sheet (M-5-5-2_Power of Ten Practice and KEY)
student copies of Lesson 2 Exit Ticket and one answer key (M-5-5-2_Lesson 2 Exit Ticket and KEY)
- optional: Power of Ten Flash Cards (M-5-5-2_Flash Cards). Print these back to back so answers will line up correctly on the back of the question cards. Cut them apart prior to the lesson and store in envelopes or zip-top baggies. This is an Extension activity.
- optional: student copies the Decimal Dash sheet (M-5-5-2_Decimal Dash and KEY). This is an Extension activity.
- optional: student copies of the Make That Negative sheet (M-5-5-2_Make That Negative and KEY). This is an Extension activity.

- Observation during the Power of Ten group activity can be used to assess class comprehension. 
- Student presentations of Powers of Ten word problems can be used to assess individual skill levels and comprehension. 
- The Lesson 2 Exit Ticket can be used to assess students' understanding of lesson contents. 

Scaffolding, Active Engagement, Modeling, and Formative Assessment 
W: The lesson aims to investigate how multiplying by 10 and powers of 10 affect the movement of the decimal point. Basic exponents will also be discussed. 
H: Students will analyze patterns that emerge when multiplying numbers by 10 and powers of 10. Base-ten blocks will also be used to engage students in meaningful investigation and discovery of multiplication by tens. 
E: Students work together to use base-ten blocks to represent various multiplication problems. Students practice grouping and creating multiple representations of the same problem. As students experiment with numerous methods to represent numbers, they build fluency with base ten and a greater understanding of place value. 
R: Students can review their learning by completing the Power of Ten practice sheet. This will also allow you to highlight any concepts from the class that require reteaching, reinforcement, or clarification before the final evaluation. 
E: Students will complete an exit ticket to demonstrate what they have learned. You can use these data to decide whether reteaching is required at a later time. 
T: To personalize the lesson to your students' requirements, refer to the Extension section for options. Use the Small Group exercises with students who could benefit from more practice. The Expansion is designed for use with students who demonstrate proficiency and want to be challenged beyond the requirements of the standard. The Routine suggestions can be used to reinforce and practice lesson content at any point during the school year. 
O: The lesson is exploratory in nature. Hands-on activity with base-ten blocks reinforces the concept of multiplying by powers of ten and allows students to visualize the process. Students detect patterns in products and quotients using powers of ten. Students construct strategies that they will share, summarize, and put into practice. An exit ticket is supplied to ensure comprehension, and rehabilitation and expansion activities are accessible as needed in the Extension section. 

As you begin the lesson, write the phrases product, quotient, and exponent on the whiteboard. 

"Today we'll start a new math lesson. Before we begin, we should study a few vocabulary words. Can someone remind us what a product is?" (The answer to a multiplication problem.) 

"Who remembers what a quotient is?" (The answer to a division question.) 

"I want you all to think about what an exponent is." 

Allow students about one minute to think. Use the fist-to-five method to assess students' comprehension of exponents.

"With a display of fingers, tell me how certain you are that you understand what an exponent is. If you are certain you know and could provide an example, raise your hand and show me all 5 fingers like this. If you do not have any idea what it is, raise your hand and show me zero fingers by closing your fist like this. If you have an idea but are not too sure, raise three or four fingers as shown. You can use any number between 0 and 5. When I give the signal, please show me your fist-to-five response."

If many students raise four or five fingers, ask a few to explain the definition and/or provide an example of an exponent in their own words. If most students seem doubtful, reassure them that they will learn it in today's class. Use this to assess how much time should be spent throughout the class on the basic definition and examples of exponents. 

"In today's lesson, we'll be focusing a lot on exponents and the number ten. We'll look at powers of ten, as well as patterns in products and quotients involving them. These patterns will allow you to rapidly and simply perform calculations with both large and small numbers."

Show the following or a similar list on the board:

724 × 10 =
724 × 100 =
724 × 1,000 =
724 ÷ 10 = 
724 ÷ 100 = 
724 ÷ 1,000 =
Allow students to utilize calculators to find the products and quotients, or use a calculator that is displayed in the classroom. Ask students to write three remarks on the problems, products, and quotients on a sheet of paper. Make a random selection of students to share their observations. Pay attention to student replies linked to patterns of zeros and decimal point movement, such as:

"When we multiplied by 1,000 it was like our product was 724 with three zeros added on the end." (Ask what happens when you multiply by 10. Explain that multiplying by 1,000 is equivalent to multiplying by 10 × 10 × 10.)
"When we divided 724 by 100 the answer was still 7, then 2, then 4, but with a decimal between the 7 and 2 (or 2 places from the end of the number)." Ask students how much the decimal has moved from its original position. They may claim that there was no decimal to begin with. Take this opportunity to point out that a decimal can be added at the end of any whole number, though we normally only show it when there is a digit in the tenth place. Ask the class to describe a relationship or pattern that could explain why the decimal moved two places to the left.

"You discovered numerous fascinating patterns in this set of calculations. These are the types of numerical relationships and patterns we will look at today as we move through our lesson." 

Hand out base-ten blocks or the paper version from lesson 1. (see M-5-5-1_Paper Base-Ten Models in the Resources folder). 

"In the problems we just discussed, you noted various patterns involving zeros and the movement of the decimal point. To depict these patterns, we'll be using base-ten blocks. Once we understand how these patterns work, we can apply them to efficiently solve numerous multiplication and division problems.

"Let's start with 1 × 10 = 10. Use base-ten blocks to represent 1 × 10." Have two students illustrate this on the board. (look at student work to select a student to demonstrate each display shown below).



"As we can see, the symbol 1 × 10 represents either one group of ten or ten groups of one. All of these are the equivalent of one long. Ten ones make one group of ten or one long. Remember that every set of ten can be regrouped to become one of the next larger place value. How would you modify your blocks to reflect 2 × 10? What about 5 × 10?"



"It looks like most of you made two groups of 10, which are equal to 20, and five groups of 10, each totaling 50. Without speaking anything aloud, consider these examples and what you notice about their answers. 

"Next show me 10 × 10 with your blocks."



"Complete the following statement with your block representation: 

10 × 10 equals _______________, or __________ flat(s)." (10 groups of 10; 1 flat) 

"When we have ten groups of ten, we must regroup into one group in the hundreds place, leaving zero tens and zero ones. 

"Show me 20 × 10 and 12 × 10."



"Let's go a little higher. Use your blocks to represent 100 × 10."



"Please complete this sentence: 100 × 10 means _________ or __________, which is equivalent to _______." (100 groups of 10 or 10 groups of 100; 1 cube or 1,000) 

"In this case, we had so many hundreds that we could regroup them to one group of a thousand leaving a zero in the hundreds, tens, and ones places." 

"Has anybody noticed a theme or pattern in what I have asked you to multiply by?" (typically by 10, 100, or 1,000) 

"How are all of the products related?" (They end with in same amount of zeros as the power of ten we multiplied by.) 

"These are what we call powers of 10." Exponents are used to express the strength of a number. In other words, an exponent represents how many times a number should be multiplied by itself. For example, consider the number 32. The base number by which we shall multiply is three. The exponent is two. This means that three will be multiplied two times, resulting in 3 × 3 = 9. How many of you have that right in your head?" A typical mistake is to multiply 3 × 2 = 6 rather than 3 × 3 = 9. 

"What do the numbers 5³, 12², and 10³ represent? Raise your hand if you know any of the answers." (5³ = 5 x 5 x 5 = 125; 12² = 12 x 12 = 144; 10³ = 10 x 10 x 10 = 1000)

"Multiplying by 100 is equivalent to multiplying by 10 and then 10 again. Another method to express this is 'multiply by 10².' This is read as 'ten to the second power' or 'ten squared'. Why ten squared? Think about your base ten blocks. The block that is 10 ones by 10 ones and contains 100 ones is called a _________." (flat, which is square).

“10² = 10 × 10 = 100.

“Fill in the blanks on this one: 10³ = ___ × ___ × ___ = _____.” (10 × 10 × 10 = 1,000)

"The expression 10³ can be translated as 'ten to the third power' or 'ten cubed.' Which base-ten block measures 10 ones by 10 ones by 10 ones?" (The cube.) "Yes! The cube represents 1,000. The term cubed is used to mark anything to the third power or to describe a three-dimensional measurement, like a cube. For the remainder of our work with powers of 10, we can use the standard form numbers 10, 100, and 1,000, or the exponent form, such as 10² and 10³." 

For the following part of the lesson, arrange students in groups of three.

"We have primarily worked on multiplication problems. It is also important to examine the patterns of division. I'd like each of your groups to divide two of the following problems using paper and pencil or base-ten blocks. Use any method you know. Each of you should solve different problems. If you don't have enough people in your group to finish each problem, do more than two so you have all of the quotients to compare. When you're done, arrange your answers in order."

[Note: If some or all of your students find long division too difficult or time-consuming, you may allow them to use a calculator. It may be more difficult to make some of the math connections with a calculator.]

6,800 ÷ 10 =                            45.68 ÷ 10 =
6,800 ÷ 100 =                          45.68 ÷ 100 =
6,800 ÷ 1,000 =                        45.68 ÷ 1,000 =
Walk around the room, helping students who are having difficulty with the division process. After the groups have finished, proceed to the next step. 

"Let's think about what you found."

Distribute three sticky notes to each group. Allow them 5–10 minutes to discuss the similarities and differences between the multiplication and division questions. Each group should use sticky notes to write down one similarity, one difference, and one pattern or method they discovered to help solve multiplication or division problems involving powers of 10. Ask each group to share their ideas. Create a place on the front board or poster paper to post each note. Group similarities, differences, and strategies together as groups continue to share. Make sure to emphasize proper thinking and strategies as they are presented. Use guiding questions to help students correct any misconceptions that arise. If any further unique ideas or misconceptions arise while observing, bring them to the attention of the group. Make sure to correct these misconceptions.

Summarize the patterns and strategies that students discovered. Observations should be similar to, but not limited to, the following: 

Multiplying by a power of 10 increases the value; dividing by a power of 10 reduces the value. 
When you multiply any whole or decimal number by 10, the decimal point moves one place to the right. 
For whole numbers, this is equivalent to adding a place holder for zero at the end of the original number.
Dividing any whole or decimal number by 10 causes the decimal point to move one place to the left. 
Multiplying any whole or decimal number by 100 (or 10²) moves the decimal point two positions to the right. For whole numbers, this is equivalent to adding two place-holding zeros to the end of the original number.
Dividing any whole or decimal number by 100 (or 10²) moves the decimal point two positions to the left.
Multiplying any whole or decimal number by 1,000 (or 10³) causes the decimal to move three places to the right. For whole numbers, this is equivalent to adding three place-holding zeros to the end of the original number.
Dividing a whole or decimal value by 1,000 (or 10³) moves the decimal point three places to the left.
Multiplying or dividing by 100 (or 10²) is equivalent to multiplying or dividing by 10 and then 10 again.
Multiplying or dividing by 1000 (or 10²) is equivalent to multiplying or dividing by 10 three times in a row.
Multiplying by any power of ten moves the decimal to the right (which may result in extra zeros being inserted at the end of the number), whereas dividing by any power of ten moves the decimal to the left. (which may remove zeros from the end of the number).

Group Activity: Power of Ten Practice

For this activity, students can work in the same three-person group or in pairs. Distribute copies of the Power of Ten practice sheet (M-5-2-2_Power of Ten Practice and KEY). Allow around 15 minutes for students to finish the questions and generate a problem. Remind students that they will present the problem they created (question 15) to the class at the end of the work period. Allow roughly 15-25 minutes for students to present the problems and solutions they wrote.

Monitor students while they are at work. In each pair or small group, ask students to summarize their opinions and findings on the zero and decimal patterns from the lesson. Make ideas or ask clarifying questions to help students who have misconceptions. During work time and presentations, encourage students to correct or adjust any work that is inaccurate.

At the end of the class, each student should fill out the exit ticket (M-5-5-2_Lesson 2 Exit Ticket and KEY). Use the results to decide which extra educational strategies may be helpful to specific students.

Extension:

Routine: Throughout the school year, identify and discuss instances of powers of ten as they occur. Ask students to bring examples of powers of ten that they observe when reading or performing calculations in math or other subjects. Have a special place in the classroom, such as a bulletin board, for displaying what students have discovered and shared with the class.

Small Groups: Use these or similar stations with students who are struggling with the idea of products and quotients with powers of ten. Students can work alone or in pairs as they progress through stations 1 and 2.

Work Station 1: Flashcards 

This activity works best with a friend. Place one or more sets of Power of Ten Flash Cards at the station (M-5-5-2_Flash Cards). Students will work in pairs, with one host and one player. The host will present the cards for the player to answer. The host will be able to view and verify the answers on the back. Any question answered incorrectly should be returned to the bottom of the pile by the host and attempted again at the end of the round. When the player is completed, students switch roles.

Work Station 2: Decimal Dash. 

Students will practice writing decimal numbers as fractions with denominators of 10, 100, or 1,000 and the reverse, as soon as possible. Place Decimal Dash sheets at each station (M-5-5-2_Decimal Dash and KEY). Every student needs a copy. Also, set a timer at the station. Students should aim to finish the sheet in less than one minute, or any time is posted at the station.

Expansion: Make That Negative 
This optional practice is ideal for students who have shown proficiency with powers of ten in products and quotients and are searching for an additional challenge beyond the requirements of the standard. Students will multiply by negative powers of ten and then rewrite the multiplication with negative powers of ten as a division by positive powers of ten. Give each student or pair of students a Make That Negative sheet (M-5-5-2_Make That Negative and KEY).